$C^{1,1}$ 正则性全局估计：穿孔域上齐次复 Hessian 方程
$C^{1,1}$ Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains
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中文总结 AI 辅助
本文通过引入复线性向量场生成的辅助函数，将全局实 Hessian 估计归结到边界，证明了穿孔域上齐次复 Hessian 方程 Green 函数的 $C^{1,1}$ 正则性，并推广至复 Monge-Ampère 情形和外部问题。
中文摘要 AI 辅助
本文证明了对于 $1\leq k<n$，与复 $k$-Hessian 算子相关的 Green 函数的 $C^{1,1}$ 正则性，并给出了多重复 Green 函数相应正则性的新证明。我们针对穿孔域上齐次复 Hessian 方程的逼近问题建立了实 Hessian 估计。主要思想是引入由复线性向量场生成的新辅助函数，将全局实 Hessian 估计归结到边界。我们还处理了复 Monge-Ampère 情形。类似的实 Hessian 估计也适用于外部问题。
英文摘要
In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.
